Down-Persistent Laplacians in Topology

• Down-persistent Laplacians are lower-component operators defined as (d*)d that capture the cycle space of the lower complex in persistent Laplacian theory.
• They quantify cycle formation in both simplicial and Hilbert complexes, offering spectral control, monotonicity, and stability under filtrations.
• Their matrix representation via (B_q^K)ᵀB_q^K enables efficient sparse implementations and bridges discrete and smooth operator-theoretic constructions.

Down-persistent Laplacians are the lower-component operators in persistent Laplacian theory for an inclusion $K\subset L$ of simplicial complexes, and more generally for inclusions $P\subset Q$ of Hilbert complexes. In the simplicial setting, for fixed $q\ge 0$, the down-persistent Laplacian is defined on the $q$-chains of the smaller complex by

$$\Delta^{K,L}_{q,\downarrow}=(\partial_q^K)^*\partial_q^K,$$

equivalently $(B_q^K)^T B_q^K$ in the canonical basis, where $B_q^K$ is the boundary matrix of $\partial_q^K$. It is therefore formally the ordinary down-Laplacian of $K$, but it is studied as one summand of the full persistent Laplacian

$$\Delta_q^{K,L}=\Delta^{K,L}_{q,\uparrow}+\Delta^{K,L}_{q,\downarrow} =\partial_{q+1}^{L,K}(\partial_{q+1}^{L,K})^*+(\partial_q^K)^*\partial_q^K.$$

Its kernel is the cycle space of the lower level, whereas the kernel of the full sum recovers persistent homology; its spectrum also admits explicit monotonicity and interleaving-stability statements under filtrations (Mémoli et al., 2020, Wolf et al., 24 Sep 2025).

1. Definitions and operator-theoretic formulations

For an inclusion $P\subset Q$0 of simplicial complexes, the ordinary chain groups are

$P\subset Q$1

with boundary operator $P\subset Q$2. With respect to the standard Euclidean inner products on these chain spaces, the down-persistent Laplacian is

$P\subset Q$3

If $P\subset Q$4 denotes the unweighted boundary matrix, then

$P\subset Q$5

In the pairwise persistent setting, the corresponding up-term is built from the restricted boundary map $P\subset Q$6, defined on those $P\subset Q$7-chains in $P\subset Q$8 whose boundary lands in $P\subset Q$9, and the full persistent Laplacian is their sum (Mémoli et al., 2020).

The same construction extends beyond simplicial complexes. For Hilbert complexes $q\ge 0$0, the general definition is

$q\ge 0$1

This operator is nonnegative and self-adjoint, with quadratic form

$q\ge 0$2

defined on $q\ge 0$3. In the de Rham setting, for compact, oriented Riemannian manifolds $q\ge 0$4, one has

$q\ge 0$5

where $q\ge 0$6 is the usual Kodaira adjoint of the exterior derivative. Thus the down-persistent Laplacian is not restricted to finite simplicial models; it persists as an operator-theoretic construction across discrete and smooth categories (Wolf et al., 24 Sep 2025).

A basic structural feature is that the down-persistent operator depends only on the lower complex. In the filtered simplicial notation $q\ge 0$7, one has

$q\ge 0$8

so the dependence on the upper level enters only through the up-component (Wei et al., 2023).

2. Algebraic-topological role

The central algebraic-topological statement for persistent Laplacians is that the number of zero eigenvalues of the full persistent Laplacian recovers the persistent Betti number: $q\ge 0$9 where $q$0. In block form, with

$q$1

the full operator is

$q$2

and the standard discrete Hodge argument yields

$q$3

The rightmost quotient is exactly the image of $q$4, hence has dimension $q$5 (Mémoli et al., 2020).

For the down-persistent operator alone, the kernel is simpler: $q$6 That is, the zero eigenspace of the down-term is the space of all $q$7-cycles at the lower scale $q$8, not yet quotiented by boundaries that appear up to time $q$9. The full persistent homology appears only after combining the down- and up-terms. In the persistent-Hodge decomposition,

$$\Delta^{K,L}_{q,\downarrow}=(\partial_q^K)^*\partial_q^K,$$0

the middle summand is the space of persistent harmonic $$\Delta^{K,L}_{q,\downarrow}=(\partial_q^K)^*\partial_q^K,$$1-chains, isomorphic to $$\Delta^{K,L}_{q,\downarrow}=(\partial_q^K)^*\partial_q^K,$$2 (Wei et al., 2023).

This distinction addresses a common misconception. The down-persistent Laplacian is not, by itself, a complete persistent-homological invariant. Its role is to identify the cycle space at the lower level, while persistent homology is recovered from the harmonic space of the full persistent Laplacian. The 2020 treatment makes this point sharply: the down-persistent Laplacian is “nothing more exotic than the ordinary ‘down’ or ‘down-Laplace’ term of $$\Delta^{K,L}_{q,\downarrow}=(\partial_q^K)^*\partial_q^K,$$3 alone,” whereas the genuinely new persistent effects enter through the up-term and the full sum (Mémoli et al., 2020).

3. Spectral structure, monotonicity, and stability

In a filtration $$\Delta^{K,L}_{q,\downarrow}=(\partial_q^K)^*\partial_q^K,$$4, one may form $$\Delta^{K,L}_{q,\downarrow}=(\partial_q^K)^*\partial_q^K,$$5 for each pair $$\Delta^{K,L}_{q,\downarrow}=(\partial_q^K)^*\partial_q^K,$$6. Because this operator depends only on $$\Delta^{K,L}_{q,\downarrow}=(\partial_q^K)^*\partial_q^K,$$7, its $$\Delta^{K,L}_{q,\downarrow}=(\partial_q^K)^*\partial_q^K,$$8-th eigenvalue $$\Delta^{K,L}_{q,\downarrow}=(\partial_q^K)^*\partial_q^K,$$9 is independent of $(B_q^K)^T B_q^K$0 and nonincreasing in $(B_q^K)^T B_q^K$1. Its spectrum is non-negative. For the full persistent Laplacian, the eigenvalues on intervals are shown by Schur-complement and interlacing arguments to be $(B_q^K)^T B_q^K$2-Lipschitz with respect to the usual interleaving distance on filtrations; in the down-case the argument is simpler precisely because the operator depends only on the lower level (Mémoli et al., 2020).

The generalized Hilbert-complex formulation sharpens this into explicit monotonicity theorems. For $(B_q^K)^T B_q^K$3, if $(B_q^K)^T B_q^K$4 denotes the $(B_q^K)^T B_q^K$5-th smallest eigenvalue of $(B_q^K)^T B_q^K$6, then

$(B_q^K)^T B_q^K$7

Under any two filtrations $(B_q^K)^T B_q^K$8 and $(B_q^K)^T B_q^K$9,

$B_q^K$0

where $B_q^K$1 is the standard interleaving distance on filtrations and on functions of intervals. The down-persistent spectra are therefore fully stable in the interleaving sense (Wolf et al., 24 Sep 2025).

A further structural result concerns the relation between the component spectra and the spectrum of the full persistent Laplacian. Writing

$B_q^K$2

the operator $B_q^K$3, restricted to the orthogonal complement of its kernel, is unitarily equivalent to the direct sum

$B_q^K$4

Hence the nonzero spectrum of the full Laplacian is the union, with multiplicities, of the nonzero spectra of its up- and down-components. This is significant because the same paper states that the classical persistent Laplacian fails the desirable properties of monotonicity and stability, while the component maps satisfy these properties individually (Wolf et al., 24 Sep 2025).

4. Matrix forms and computation

In orthonormal bases for $B_q^K$5 and $B_q^K$6, if $B_q^K$7, then $B_q^K$8 and

$B_q^K$9

In the discrete simplicial case, this is exactly the down-Laplacian, sometimes described as the Kirchhoff matrix on $\partial_q^K$0-simplices. In the de Rham case, an entrywise matrix representation depends on a choice of basis of forms and is typically infinite-dimensional, but the abstract operator remains $\partial_q^K$1 (Wolf et al., 24 Sep 2025).

For simplicial complexes, one direct algorithm is to assemble $\partial_q^K$2 from the sparse boundary matrix $\partial_q^K$3. The stated cost is $\partial_q^K$4, or $\partial_q^K$5 in the worst case, with practical behavior $\partial_q^K$6. An equivalent combinatorial form writes

$\partial_q^K$7

where $\partial_q^K$8 is the number of $\partial_q^K$9-faces of the $K$0-simplex $K$1, and $K$2 if $K$3 share a common $K$4-face and $K$5 otherwise. Both constructions run in $K$6 by scanning pairs of simplices or faces of each simplex (Mémoli et al., 2020).

At the level of practical eigenspectrum computation, the standard workflow at a single scale $K$7 is: enumerate all $K$8- and $K$9-simplices, assemble the signed incidence matrix $$\Delta_q^{K,L}=\Delta^{K,L}_{q,\uparrow}+\Delta^{K,L}_{q,\downarrow} =\partial_{q+1}^{L,K}(\partial_{q+1}^{L,K})^*+(\partial_q^K)^*\partial_q^K.$$0, form the sparse matrix $$\Delta_q^{K,L}=\Delta^{K,L}_{q,\uparrow}+\Delta^{K,L}_{q,\downarrow} =\partial_{q+1}^{L,K}(\partial_{q+1}^{L,K})^*+(\partial_q^K)^*\partial_q^K.$$1, and solve for the $$\Delta_q^{K,L}=\Delta^{K,L}_{q,\uparrow}+\Delta^{K,L}_{q,\downarrow} =\partial_{q+1}^{L,K}(\partial_{q+1}^{L,K})^*+(\partial_q^K)^*\partial_q^K.$$2 smallest eigenpairs, for example via Lanczos or ARPACK. The survey states that the complexity is dominated by assembling $$\Delta_q^{K,L}=\Delta^{K,L}_{q,\uparrow}+\Delta^{K,L}_{q,\downarrow} =\partial_{q+1}^{L,K}(\partial_{q+1}^{L,K})^*+(\partial_q^K)^*\partial_q^K.$$3, linear in the number of simplices, and by the iterative eigensolve, approximately $$\Delta_q^{K,L}=\Delta^{K,L}_{q,\uparrow}+\Delta^{K,L}_{q,\downarrow} =\partial_{q+1}^{L,K}(\partial_{q+1}^{L,K})^*+(\partial_q^K)^*\partial_q^K.$$4. It also emphasizes that in practice one rarely needs the full spectrum, but rather the smallest few eigenvalues and eigenvectors (Wei et al., 2023).

5. Examples and low-dimensional behavior

A finite example given in the generalized treatment takes $$\Delta_q^{K,L}=\Delta^{K,L}_{q,\uparrow}+\Delta^{K,L}_{q,\downarrow} =\partial_{q+1}^{L,K}(\partial_{q+1}^{L,K})^*+(\partial_q^K)^*\partial_q^K.$$5 as the square with boundary edges embedded into the same square with the diagonal $$\Delta_q^{K,L}=\Delta^{K,L}_{q,\uparrow}+\Delta^{K,L}_{q,\downarrow} =\partial_{q+1}^{L,K}(\partial_{q+1}^{L,K})^*+(\partial_q^K)^*\partial_q^K.$$6 added. If the oriented edges in $$\Delta_q^{K,L}=\Delta^{K,L}_{q,\uparrow}+\Delta^{K,L}_{q,\downarrow} =\partial_{q+1}^{L,K}(\partial_{q+1}^{L,K})^*+(\partial_q^K)^*\partial_q^K.$$7 are $$\Delta_q^{K,L}=\Delta^{K,L}_{q,\uparrow}+\Delta^{K,L}_{q,\downarrow} =\partial_{q+1}^{L,K}(\partial_{q+1}^{L,K})^*+(\partial_q^K)^*\partial_q^K.$$8, $$\Delta_q^{K,L}=\Delta^{K,L}_{q,\uparrow}+\Delta^{K,L}_{q,\downarrow} =\partial_{q+1}^{L,K}(\partial_{q+1}^{L,K})^*+(\partial_q^K)^*\partial_q^K.$$9, $P\subset Q$00, $P\subset Q$01, then

$P\subset Q$02

with eigenvalues $P\subset Q$03. Its kernel is exactly the cycle space generated by $P\subset Q$04. The same matrix persists after enlarging to $P\subset Q$05 because the down-persistent definition depends only on $P\subset Q$06 (Wolf et al., 24 Sep 2025).

A second example is the filled 2-simplex on vertices $P\subset Q$07. For $P\subset Q$08, the down-Laplacian

$P\subset Q$09

has eigenvalues $P\subset Q$10. The single zero eigenvalue reflects one 1-cycle; if a filtration is considered in which the three edges appear at scale $P\subset Q$11 and the interior 2-simplex appears at scale $P\subset Q$12, then the down-piece at $P\subset Q$13 is unchanged, while the up-piece is what kills the 1-cycle at time $P\subset Q$14 (Wei et al., 2023).

At $P\subset Q$15, the pairwise homological definition gives

$P\subset Q$16

because $P\subset Q$17. Accordingly, the 2020 analysis states that nothing new arises from the down-term at $P\subset Q$18. In the graph case, the novel persistent phenomena such as effective-resistance preservation and persistent Cheeger inequalities appear only after incorporating the up-term and, in the Schur-complement formulation, interpreting the up persistent Laplacian as a Kron reduction of $P\subset Q$19 to $P\subset Q$20 (Mémoli et al., 2020).

For $P\subset Q$21, the same source defines an analogue of effective resistance on $P\subset Q$22-simplices, or more generally on $P\subset Q$23-dimensional current generators: $P\subset Q$24 where $P\subset Q$25 is the boundary of a chosen current generator $P\subset Q$26. In the 1-dimensional case this recovers the usual vertex-vertex resistance, but the down-term alone does not connect two different levels $P\subset Q$27. The new persistence resistance emerges only when the up-term is added and a Schur complement is performed (Mémoli et al., 2020).

6. Extensions, interpretations, and conceptual scope

The survey on persistent topological Laplacians emphasizes that the same down construction carries over whenever one has a chain complex $P\subset Q$28 of inner-product spaces together with an inclusion of chain complexes. In that general setting, the down-persistent Laplacian is simply $P\subset Q$29. The survey lists path complexes and digraphs, flag complexes, hypergraphs and hyperdigraphs, cellular cosheaves and sheaves, and $P\subset Q$30-chain complexes as settings where this pattern reappears, with the appropriate boundary or coboundary replacing the simplicial boundary (Wei et al., 2023).

Within this broader landscape, the down-persistent operator retains a sharply delimited meaning. The survey describes its sole role as tracking which $P\subset Q$31-chains have zero boundary at a given scale. Its zero eigenspace is the cycle space; its positive eigenvalues are described as measuring how costly it is to create a $P\subset Q$32-chain whose boundary vanishes, and as reflecting clustering and connectivity of $P\subset Q$33-simplices as well as higher-order bottlenecks. This suggests a division of labor within persistent Laplacian theory: the down-part provides the lower-level cycle geometry, while the up-part governs the interaction with later boundaries (Wei et al., 2023).

The 2020 and 2025 works both reinforce this division. The 2020 analysis concludes that “all of the genuinely persistent new phenomena—Schur-complement formulas, Kron reduction, effective-resistance preservation, persistent Cheeger—reside in the up-persistent term and its interaction with the down term only through their sum.” The 2025 framework, however, shows that the down- and up-components are not merely auxiliary: each is individually monotone and stable, and together their nonzero spectra completely determine the nonzero spectrum of the full persistent Laplacian (Mémoli et al., 2020, Wolf et al., 24 Sep 2025).

For that reason, down-persistent Laplacians occupy a dual position in the theory. On one hand they are the simplest part of the persistent operator, identical in form to the ordinary down-Laplacian of the lower complex. On the other hand, they are mathematically robust invariants in their own right, admitting operator-theoretic generalization, explicit spectral control under refinement and interleaving, and efficient sparse-matrix implementations across a range of topological settings (Wolf et al., 24 Sep 2025).